If $X\subset L^1$ is a closed vector space and $X\subset \bigcup_{1<p\leq\infty} L^p$ then $X\subset L^q$ for some $q>1$.

Closedness of $X_n$'s follows immediately from Fatou's Lemma: If $\|f_j\|_{1+\frac 1 n} \leq n$ for all $j$ and $f_j \to f$ in $X$ then there is a subsequence $f_{j_i}$ which converges a.e. so $\|f\|_{1+\frac 1 n} \leq \lim \inf_k \|f_{j_k}\|_{1+\frac 1 n} \leq n$.

Suppose there is an open ball $B(f_0,r)$ in $X$ which is $ \subset X_{n_0}$. Let $f \in X$. Then $\|f_0+\frac f N\|_{1+\frac 1 {n_0}} \leq n_0$ for $N$ sufficiently large (since $f_0+\frac f N \in B(f_0,r)$). This implies that $f_0+\frac f N\in L^{1+\frac 1 {n_0}}(\Omega)$ and we also have $f_0 \in L^{1+\frac 1 {n_0}}(\Omega)$. Since $L^{1+\frac 1 {n_0}}(\Omega)$ is a vector space it follows that $f \in L^{1+\frac 1 {n_0}}(\Omega)$.